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53edo

Revision as of 14:23, 20 August 2022 by FloraC (talk | contribs) (-monotonicity -major 2nd in infobox)
← 52edo 53edo 54edo →
Prime factorization 53 (prime)
Step size 22.6415 ¢ 
Fifth 31\53 (701.887 ¢)
(convergent)
Semitones (A1:m2) 5:4 (113.2 ¢ : 90.57 ¢)
Consistency limit 9
Distinct consistency limit 9

The famous 53 equal divisions of the octave (53edo), or 53(-tone) equal temperament (53tet, 53et) when viewed from a regular temperament perspective, divides the octave into 53 equal comma-sized parts of around 22.6 cents each.

English Wikipedia has an article on:

Theory

53edo is notable as a 5-limit system, a fact apparently first noted by Isaac Newton, notably tempering out the schisma (32805/32768), the kleisma (15625/15552), the amity comma (1600000/1594323), the semicomma (2109375/2097152), and the vulture comma (10485760000/10460353203). In the 7-limit it tempers out 225/224, 1728/1715 and 3125/3087, the marvel comma, the gariboh, and the orwell comma. In the 11-limit, it tempers out 99/98 and 121/120 (in addition to their difference, 540/539), and is the optimal patent val for Big Brother temperament, which tempers out both, as well as 11-limit orwell temperament, which also tempers out the 11-limit commas 176/175 and 385/384. In the 13-limit, it tempers out 169/168, 275/273, 325/324, 625/624, 676/675, 1001/1000, and 2080/2079, and gives the optimal patent val for athene temperament. It is the seventh strict zeta edo. It can also be treated as a no-elevens, no-seventeens tuning, on which it is consistent all the way up to the 23-odd-limit.

53edo has also found a certain dissemination as an EDO tuning for Arabic, Turkish, Persian music.

53edo is the 16th prime edo, following 47edo and coming before 59edo.

Prime harmonics

Approximation of prime harmonics in 53edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.07 -1.41 +4.76 -7.92 -2.79 +8.25 -3.17 +5.69 -10.71 +9.68
Relative (%) +0.0 -0.3 -6.2 +21.0 -35.0 -12.3 +36.4 -14.0 +25.1 -47.3 +42.8
Steps
(reduced) 		53
(0) 		84
(31) 		123
(17) 		149
(43) 		183
(24) 		196
(37) 		217
(5) 		225
(13) 		240
(28) 		257
(45) 		263
(51)

Intervals

# Solfege Cents Approximate ratios* Ups and downs notation
0 do 0.00 1/1 P1 unison D
1 di 22.64 81/80, 64/63, 50/49 ^1 up unison ^D
2 daw 45.28 49/48, 36/35, 33/32, 128/125 ^^1,
vvm2 		double-up unison,
double-down minor 2nd 		^^D,
vvEb
3 ro 67.92 25/24, 28/27, 22/21, 27/26, 26/25 vm2 downminor 2nd vEb
4 rih 90.57 19/18, 20/19, 21/20, 256/243 m2 minor 2nd Eb
5 ra 113.21 16/15, 15/14 ^m2 upminor 2nd ^Eb
6 ru 135.85 14/13, 13/12, 27/25 v~2 downmid 2nd ^^Eb
7 ruh 158.49 35/32, 12/11, 11/10, 57/52, 800/729 ^~2 upmid 2nd vvE
8 reh 181.13 10/9 vM2 downmajor 2nd vE
9 re 203.77 9/8 M2 major 2nd E
10 ri 226.42 8/7, 256/225 ^M2 upmajor 2nd ^E
11 raw 249.06 15/13, 144/125 ^^M2,
vvm3 		double-up major 2nd,
double-down minor 3rd 		^^E,
vvF
12 ma 271.70 7/6, 75/64 vm3 downminor 3rd vF
13 meh 294.34 13/11, 19/16, 32/27 m3 minor 3rd F
14 me 316.98 6/5 ^m3 upminor 3rd ^F
15 mu 339.62 11/9, 243/200 v~3 downmid 3rd ^^F
16 muh 362.26 16/13, 100/81 ^~3 upmid 3rd vvF#
17 mi 384.91 5/4 vM3 downmajor 3rd vF#
18 maa 407.55 19/15, 24/19, 81/64 M3 major 3rd F#
19 mo 430.19 9/7, 14/11 ^M3 upmajor 3rd ^F#
20 maw 452.83 13/10, 125/96 ^^M3,
vv4 		double-up major 3rd,
double-down 4th 		^^F#,
vvG
21 fe 475.47 21/16, 25/19, 675/512, 320/243 v4 down 4th vG
22 fa 498.11 4/3 P4 perfect 4th G
23 fih 520.75 27/20 ^4 up 4th ^G
24 fu 543.40 11/8, 15/11, 26/19 v~4 downmid 4th ^^G
25 fuh 566.04 18/13 ^~4,
vd5 		upmid 4th,
downdim 5th 		vvG#,
vAb
26 fi 588.68 7/5, 45/32 vA4,
d5 		downaug 4th,
dim 5th 		vG#,
Ab
27 se 611.32 10/7, 64/45 A4,
^d5 		aug 4th,
updim 5th 		G#,
^Ab
28 suh 633.96 13/9 ^A4,
v~5 		upaug 4th,
downmid 5th 		^G#,
^^Ab
29 su 656.60 16/11, 19/13, 22/15 ^~5 upmid 5th vvA
30 sih 679.25 40/27 v5 down 5th vA
31 sol 701.89 3/2 P5 perfect 5th A
32 si 724.53 32/21, 38/25, 243/160, 1024/675 ^5 up 5th ^A
33 saw 747.17 20/13, 192/125 ^^5,
vvm6 		double-up 5th,
double-down minor 6th 		^^A,
vvBb
34 lo 769.81 14/9, 25/16, 11/7 vm6 downminor 6th vBb
35 leh 792.45 19/12, 30/19, 128/81 m6 minor 6th Bb
36 le 815.09 8/5 ^m6 upminor 6th ^Bb
37 lu 837.74 13/8, 81/50 v~6 downmid 6th ^^Bb
38 luh 860.38 18/11, 400/243 ^~6 upmid 6th vvB
39 la 883.02 5/3 vM6 downmajor 6th vB
40 laa 905.66 22/13, 27/16, 32/19 M6 major 6th B
41 lo 928.30 12/7 ^M6 upmajor 6th ^B
42 law 950.94 26/15, 125/72 ^^M6,
vvm7 		double-up major 6th,
double-down minor 7th 		^^B,
vvC
43 ta 973.58 7/4 vm7 downminor 7th vC
44 teh 996.23 16/9 m7 minor 7th C
45 te 1018.87 9/5 ^m7 upminor 7th ^C
46 tu 1041.51 64/35, 11/6, 20/11, 729/400 v~7 downmid 7th ^^C
47 tuh 1064.15 13/7, 24/13, 50/27 ^~7 upmid 7th vvC#
48 ti 1086.79 15/8 vM7 downmajor 7th vC#
49 tih 1109.43 19/10, 36/19, 40/21, 243/128 M7 major 7th C#
50 to 1132.08 48/25, 27/14, 21/11, 52/27, 25/13 ^M7 upmajor 7th ^C#
51 taw 1154.72 96/49, 35/18, 64/33, 125/64 ^^M7,
vv8 		double-up major 7th,
double-down 8ve 		^^C#,
vvD
52 da 1177.36 160/81, 63/32, 49/25 v8 down 8ve vD
53 do 1200.00 2/1 P8 perfect 8ve D

* Based on interpreting 53edo as a no-17's 19-limit temperament. Italics represent inconsistent intervals which are mapped by the 19-limit patent val to their second-best (as opposed to best) approximation in 53edo.

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

quality color monzo format examples
downminor zo (a, b, 0, 1) 7/6, 7/4
minor fourthward wa (a, b) with b < -1 32/27, 16/9
upminor gu (a, b, -1) 6/5, 9/5
downmid ilo (a, b, 0, 0, 1) 11/9, 11/6
upmid lu (a, b, 0, 0, -1) 12/11, 18/11
downmajor yo (a, b, 1) 5/4, 5/3
major fifthward wa (a, b) with b > 1 9/8, 27/16
upmajor ru (a, b, 0, -1) 9/7, 12/7

All 53edo chords can be named using ups and downs. An up, down or mid after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are.

Here are the zo, gu, ilo, lu, yo and ru triads:

color of the 3rd JI chord notes as edosteps notes of C chord written name spoken name
zo 6:7:9 0-12-31 C vEb G Cvm C downminor
gu 10:12:15 0-14-31 C ^Eb G C^m C upminor
ilo 18:22:27 0-15-31 C ^^Eb G Cv~ C downmid
lu 22:27:33 0-16-31 C vvE G C^~ C upmid
yo 4:5:6 0-17-31 C vE G Cv C downmajor or C down
ru 14:18:21 0-19-31 C ^E G C^ C upmajor or C up

For a more complete list, see Ups and Downs Notation #Chords and Chord Progressions.

Notation

Sagittal

The following table shows sagittal notation accidentals in one apotome for 53edo.

Steps 0 1 2 3 4 5
Symbol            

Relationship to 12edo

Whereas 12edo has a circle of twelve 5ths, 53edo has a spiral of twelve 5ths (since 31\53 is on the 7\12 kite in the scale tree). This shows 53edo in a 12edo-friendly format. Excellent for introducing 53edo to musicians unfamiliar with microtonal music. The two innermost and two outermost intervals on the spiral are duplicates.

 

JI approximation

53edo provides excellent approximations for the classic 5-limit just chords and scales, such as the Ptolemy-Zarlino "just major" scale.

Interval Ratio Size Difference
Perfect fifth 3/2 31 −0.07 cents
Major third 5/4 17 −1.40 cents
Minor third 6/5 14 +1.34 cents
Major tone 9/8 9 −0.14 cents
Minor tone 10/9 8 −1.27 cents
Diat. semitone 16/15 5 +1.48 cents

One notable property of 53edo is that it offers good approximations for both just and Pythagorean major thirds.

The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO is practically equal to an extended Pythagorean. The 14- and 17- degree intervals are also very close to 6/5 and 5/4 respectively, and so 5-limit tuning can also be closely approximated. In addition, the 43-degree interval is only 4.8 cents away from the just ratio 7/4, so 53edo can also be used for 7-limit harmony, tempering out the septimal kleisma, 225/224.

15-odd-limit interval mappings

The following table shows how 15-odd-limit intervals are represented in 53edo. Octave-reduced prime harmonics are bolded; inconsistent intervals are in italic.

15-odd-limit intervals by direct mapping (even if inconsistent)
Interval, complement Error (abs, ¢) Error (rel, %)
3/2, 4/3 0.068 0.3
9/8, 16/9 0.136 0.6
9/5, 10/9 1.272 5.6
15/13, 26/15 1.316 5.8
5/3, 6/5 1.340 5.9
13/10, 20/13 1.384 6.1
5/4, 8/5 1.408 6.2
15/8, 16/15 1.476 6.5
13/9, 18/13 2.655 11.7
13/12, 24/13 2.724 12.0
13/8, 16/13 2.792 12.3
7/4, 8/7 4.759 21.0
7/6, 12/7 4.827 21.3
9/7, 14/9 4.895 21.6
13/11, 22/13 5.130 22.7
7/5, 10/7 6.167 27.2
15/14, 28/15 6.235 27.5
15/11, 22/15 6.445 28.5
11/10, 20/11 6.514 28.8
13/7, 14/13 7.551 33.3
11/9, 18/11 7.785 34.4
11/6, 12/11 7.854 34.7
11/8, 16/11 7.922 35.0
11/7, 14/11 9.961 44.0

The following tables show how 15-odd-limit intervals are represented in 53edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 53edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
3/2, 4/3 0.068 0.3
9/8, 16/9 0.136 0.6
9/5, 10/9 1.272 5.6
15/13, 26/15 1.316 5.8
5/3, 6/5 1.340 5.9
13/10, 20/13 1.384 6.1
5/4, 8/5 1.408 6.2
15/8, 16/15 1.476 6.5
13/9, 18/13 2.655 11.7
13/12, 24/13 2.724 12.0
13/8, 16/13 2.792 12.3
7/4, 8/7 4.759 21.0
7/6, 12/7 4.827 21.3
9/7, 14/9 4.895 21.6
13/11, 22/13 5.130 22.7
7/5, 10/7 6.167 27.2
15/14, 28/15 6.235 27.5
15/11, 22/15 6.445 28.5
11/10, 20/11 6.514 28.8
13/7, 14/13 7.551 33.3
11/9, 18/11 7.785 34.4
11/6, 12/11 7.854 34.7
11/8, 16/11 7.922 35.0
11/7, 14/11 9.961 44.0
15-odd-limit intervals in 53edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
3/2, 4/3 0.068 0.3
9/8, 16/9 0.136 0.6
9/5, 10/9 1.272 5.6
15/13, 26/15 1.316 5.8
5/3, 6/5 1.340 5.9
13/10, 20/13 1.384 6.1
5/4, 8/5 1.408 6.2
15/8, 16/15 1.476 6.5
13/9, 18/13 2.655 11.7
13/12, 24/13 2.724 12.0
13/8, 16/13 2.792 12.3
7/4, 8/7 4.759 21.0
7/6, 12/7 4.827 21.3
9/7, 14/9 4.895 21.6
13/11, 22/13 5.130 22.7
7/5, 10/7 6.167 27.2
15/14, 28/15 6.235 27.5
15/11, 22/15 6.445 28.5
11/10, 20/11 6.514 28.8
13/7, 14/13 7.551 33.3
11/9, 18/11 7.785 34.4
11/6, 12/11 7.854 34.7
11/8, 16/11 7.922 35.0
11/7, 14/11 12.681 56.0

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-84 53⟩ [⟨53 84]] +0.022 0.022 0.10
2.3.5 15625/15552, 32805/32768 [⟨53 84 123]] +0.216 0.276 1.22
2.3.5.7 225/224, 1728/1715, 3125/3087 [⟨53 84 123 149]] -0.262 0.861 3.81
2.3.5.7.11 99/98, 121/120, 176/175, 2200/2187 [⟨53 84 123 149 183]] +0.248 1.279 5.64
2.3.5.7.11.13 99/98, 121/120, 169/168, 176/175, 275/273 [⟨53 84 123 149 183 196]] +0.332 1.183 5.22
2.3.5.7.11.13.19 99/98, 121/120, 169/168, 176/175, 209/208, 275/273 [⟨53 84 123 149 183 196 225]] +0.391 1.105 4.88

53et is lower in relative error than any previous equal temperaments in the 3-, 5-, and 13-limit. The next ETs doing better in these subgroups are 306, 118, and 58, respectively. It is even more prominent in the 2.3.5.7.13.19 and 2.3.5.7.13.19.23 subgroups, and the next ET doing better in either subgroup is 130.

Linear temperaments

Table of rank-2 temperaments by generator
Periods
per Octave 		Generator Cents Associated
Ratio 		Temperament
1 2\53 45.28 36/35 Quartonic
1 5\53 113.21 16/15 Misneb
1 7\53 158.49 11/10 Hemikleismic
1 9\53 203.77 9/8 Baldy
1 10\53 226.42 8/7 Semaja
1 11\53 249.06 15/13 Hemischis / hemigari
1 12\53 271.70 7/6 Orson / orwell
1 13\53 294.34 25/21 Kleiboh
1 14\53 316.98 6/5 Hanson / catakleismic / countercata
1 15\53 339.62 11/9 Amity / houborizic
1 16\53 362.26 16/13 Submajor
1 18\53 407.55 1225/972 Ditonic / coditone
1 19\53 430.19 9/7 Hamity
1 21\53 475.47 21/16 Vulture / buzzard
1 22\53 498.11 4/3 Helmholtz / garibaldi / pontiac
1 25\53 566.04 18/13 Tricot
1 26\53 588.68 45/32 Untriton / aufo

Music

See also: Category:53edo tracks

See also

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